The spike index \(\gamma \)  is the only free parameter in this analysis. For a spike of collisionless dark matter that forms about an adiabatically growing black hole, we have \(\gamma =2.25-2.5\)  [1]}, [2]}. However, if gravitational scattering of stars is important, the stellar heating effect would drive the value of \(\gamma \)  down to a minimum value \(\gamma =1.5\)  [3]}, [4]}. Such a change in the spike index depends on the heating time scale, which is given by [3]}:
\(t_{\rm heat}&=&\frac{\sqrt{3\pi } \Gamma (0.5)M_{\rm BH}}{18m \ln \Lambda } \left(\frac{GM_{\rm BH}}{r_{\rm in}^3} \right)^{-1/2}=1.2 \times 10^{15}~{\rm s} \nonumber \\&&\times \left(\frac{M_{\rm BH}}{5M_{\odot }} \right)^{1/2}\left(\frac{r_{\rm in}}{5~\rm pc} \right)^{3/2} \left(\frac{m}{M_{\odot }} \right)^{-1} \left(\frac{\ln \Lambda }{3} \right)^{-1},\) 
